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This work addresses the algorithmic tractability of hard combinatorial problems. Basically, we are considering \NP-hard problems. For those problems we can not find a polynomial time algorithm. Several algorithmic approaches already exist which deal with this dilemma. Among them we find (randomized) approximation algorithms and heuristics. Even though in practice they often work in reasonable time they usually do not return an optimal solution. If we constrain optimality then there are only two methods which suffice for this purpose: exponential time algorithms and parameterized algorithms. In the first approach we seek to design algorithms consuming exponentially many steps who are more clever than some trivial algorithm (who simply enumerates all solution candidates). Typically, the naive enumerative approach yields an algorithm with run time $\Oh^*(2^n)$. So, the general task is to construct algorithms obeying a run time of the form $\Oh^*(c^n)$ where $c<2$. The second approach considers an additional parameter $k$ besides the input size $n$. This parameter should provide more information about the problem and cover a typical characteristic. The standard parameterization is to see $k$ as an upper (lower, resp.) bound on the solution size in case of a minimization (maximization, resp.) problem. Then a parameterized algorithm should solve the problem in time $f(k)\cdot n^\beta$ where $\beta$ is a constant and $f$ is independent of $n$. In principle this method aims to restrict the combinatorial difficulty of the problem to the parameter $k$ (if possible). The basic hypothesis is that $k$ is small with respect to the overall input size. In both fields a frequent standard technique is the design of branching algorithms. These algorithms solve the problem by traversing the solution space in a clever way. They frequently select an entity of the input and create two new subproblems, one where this entity is considered as part of the future solution and another one where it is excluded from it. Then in both cases by fixing this entity possibly other entities will be fixed. If so then the traversed number of possible solution is smaller than the whole solution space. The visited solutions can be arranged like a search tree. To estimate the run time of such algorithms there is need for a method to obtain tight upper bounds on the size of the search trees. In the field of exponential time algorithms a powerful technique called Measure&Conquer has been developed for this purpose. It has been applied successfully to many problems, especially to problems where other algorithmic attacks could not break the trivial run time upper bound. On the other hand in the field of parameterized algorithms Measure&Conquer is almost not known. This piece of work will present examples where this technique can be used in this field. It also will point out what differences have to be made in order to successfully apply the technique. Further, exponential time algorithms for hard problems where Measure&Conquer is applied are presented. Another aspect is that a formalization (and generalization) of the notion of a search tree is given. It is shown that for certain problems such a formalization is extremely useful.
The subject of this thesis is a homological approach to the splitting theory of PLS-spaces, i.e. to the question for which topologically exact short sequences 0->X->Y->Z->0 of PLS-spaces X,Y,Z the right-hand map admits a right inverse. We show that the category (PLS) of PLS-spaces and continuous linear maps is an additive category in which every morphism admits a kernel and a cokernel, i.e. it is pre-abelian. However, we also show that it is neither quasi-abelian nor semi-abelian. As a foundation for our homological constructions we show the more general result that every pre-abelian category admits a largest exact structure in the sense of Quillen. In the pre-abelian category (PLS) this exact structure consists precisely of the topologically exact short sequences of PLS-spaces. Using a construction of Ext-functors due to Yoneda, we show that one can define for each PLS-space A and every natural number k the k-th abelian-group valued covariant and contravariant Ext-functors acting on the category (PLS) of PLS-spaces, which induce for every topologically exact short sequence of PLS-spaces a long exact sequence of abelian groups and group morphisms. These functors are studied in detail and we establish a connection between the Ext-functors of PLS-spaces and the Ext-functors for LS-spaces. Through this connection we arrive at an analogue of a result for Fréchet spaces which connects the first derived functor of the projective limit with the first Ext-functor and also gives sufficient conditions for the vanishing of the higher Ext-functors. Finally, we show that Ext^k(E,F) = 0 for a k greater or equal than 1, whenever E is a closed subspace and F is a Hausdorff-quotient of the space of distributions, which generalizes a result of Wengenroth that is itself a generalization of results due to Domanski and Vogt.
Das Studium des christlichen Briefstils im spätantiken Ägypten steht im Zentrum dieser Arbeit. Anhand der dokumentarischen Papyrusbriefe berührt diese Arbeit hauptsächlich die Epistolographie des Altertums, besonders die Phraseologie des frühen Christentums. So sieht man in der Arbeit die Beobachtung, Analyse und Erklärung der griechischen christlichen Privatbriefe im Hinblick auf ihre Struktur und ihre typischen christlichen Elemente insbesondere in Anredeformen, Grußformeln und typischen Redewendungen. Für diese Forschung wurden etwa 200 christliche Briefe zur intensiven Auswertung als Textgrundlage ausgewählt; zum Vergleich wurden auch heidnische Briefe aus ptolemäischer und römischer Zeit herangezogen. Diese Studie, besonders bei der Untersuchung der Formeln, bezieht sich auf den Zeitraum, zwischen dem 3. und 4./5. Jh. n.Chr.